Are there left hand cartridges for Delta two handle faucets

Looks, and sounds like, the actual metal part of the valve seat is probably messed up from over tightening. It could possibly be fixed with a grinding stone that is fine grit and the same diameter, to reface the metal. But be careful to not take too much off or the seals won't seat properly. You might also look at the warranty that came with the faucet. It could have a lifetime warranty and be as easy as getting a replacement.

10k btu should cool a fairly large room but 2 rooms maybe not, I found that the problem with these units is the filters get plugged up and removed then the coils get plugged up with dust and they don’t work as well.

A 20 degree delta +-2 is a good rule of thumb for a clean unit and if your unit has a txv or thermal expansion valve sometimes a slight adjustment will allow for a larger delta. It could have gotten bumped when it was serviced or cleaned by others the bulb in solid contact turns off the flow of refrigerant two far away two much flow the coil will ice up (this happens when really dirty also).

Can you shine a flashlight behind the coils and see the light if you open the filter housing? Or looking through if no light passes it could use a cleaning.

A can of aerosol coil cleaner is less than 10$ at most big box stores a can may be enough for your unit. (The can may show a window ac that’s ok your unit is really a very high end window unit put in the wall) .

Don’t be tempted to use vinegar acids etch the fins and eat the galvanized coating then they start rust. I think foaming cleaners are the best they push the dirt and dust out. If there is or was a smoker in the house get 2 cans smokers really plug the evaporator coils but they clean up really well.

Since the unit was recently serviced I would expect it to be in good shape but a clean unit will cool much better than a dirty unit. As we hit temp extremes the delta may drop a bit but should be 20 for most systems and can be much higher but this brings other issues like freezing up excessive condensation on the discharge side.